Is sec(t)-sin(t)tan(t)=cos(t) ?

The answer to whether sec(t)-sin(t)tan(t)=cos(t) is True

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prove sec(t)-sin(t)tan(t)=cos(t)

prove

sec (t) - sin (t) tan (t) = cos (t)

True

Solution steps

sec (t) - sin (t) tan (t) = cos (t)

Manipulating left side

sec (t) - sin (t) tan (t)

Express with sin, cos

sec (t) - sin (t) tan (t)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( t )} - sin (t) tan (t)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1}{cos ( t )} - sin (t) \frac{sin ( t )}{cos ( t )}

Simplify

\frac{1}{cos ( t )} - sin (t) \frac{sin ( t )}{cos ( t )} : \frac{1 - sin ^{2} ( t )}{cos ( t )}

\frac{1}{cos ( t )} - sin (t) \frac{sin ( t )}{cos ( t )}

sin (t) \frac{sin ( t )}{cos ( t )} = \frac{sin ^{2} ( t )}{cos ( t )}

sin (t) \frac{sin ( t )}{cos ( t )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( t ) sin ( t )}{cos ( t )}

sin (t) sin (t) = sin^{2} (t)

sin (t) sin (t)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (t) sin (t) = sin^{1 + 1} (t)

= sin^{1 + 1} (t)

Add the numbers:

1 + 1 = 2

= sin^{2} (t)

= \frac{sin ^{2} ( t )}{cos ( t )}

= \frac{1}{cos ( t )} - \frac{sin ^{2} ( t )}{cos ( t )}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - sin ^{2} ( t )}{cos ( t )}

= \frac{1 - sin ^{2} ( t )}{cos ( t )}

= \frac{1 - sin ^{2} ( t )}{cos ( t )}

Rewrite using trig identities

\frac{1 - sin ^{2} ( t )}{cos ( t )}

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

1 - sin^{2} (x) = cos^{2} (x)

= \frac{cos ^{2} ( t )}{cos ( t )}

Cancel the common factor:

cos (t)

= cos (t)

= cos (t)

We showed that the two sides could take the same form

\Rightarrow True

\frac{cos ( θ ) + cot ( θ )}{csc ( θ ) + 1} = cos (θ)

\frac{csc ^{2} ( θ ) - 2}{csc ^{2} ( θ )} = cos (2 θ)

cos^{4} (w) + 1 - sin^{4} (w) = 2 cos^{2} (w)

sin (- x) cos (\frac{π}{2} - x) = \frac{cos ( 2 x ) - 1}{2}

cos (b) + sin (b) tan (b) = sec (b)

Is sec(t)-sin(t)tan(t)=cos(t) ?
The answer to whether sec(t)-sin(t)tan(t)=cos(t) is True